# Universal Learning of Nonlinear Dynamics

arXiv: https://arxiv.org/abs/2508.11990 · HTML: https://mlsanity.com/paper/2508.11990/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 16 Aug 2025.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [The Nonstochastic Control Problem](https://mlsanity.com/paper/1911.12178/) (2019)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [Spectral Filtering for General Linear Dynamical Systems](https://mlsanity.com/paper/1802.03981/) (2018)
- [Online Learning for Time Series Prediction](https://mlsanity.com/paper/1302.6927/) (2013)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Spectral State Space Models](https://mlsanity.com/paper/2312.06837/) (2023)
- [No-Regret Prediction in Marginally Stable Systems](https://mlsanity.com/paper/2002.02064/) (2020)
- [On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters](https://mlsanity.com/paper/1809.05870/) (2018)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Theory and Algorithms for Forecasting Time Series](https://mlsanity.com/paper/1803.05814/) (2018)
- [Flash STU: Fast Spectral Transform Units](https://mlsanity.com/paper/2409.10489/) (2024)
- [Luenberger observers for discrete-time nonlinear systems](https://mlsanity.com/paper/2002.02674/) (2020)

## Built on by (most cited first, 5 of 5)

- [Research Program: Theory of Learning in Dynamical Systems](https://mlsanity.com/paper/2512.19410/) (2025)
- [The Power of Second Order Methods for Sequence Preconditioning](https://mlsanity.com/paper/2605.08390/) (2026)
- [SFO: Learning PDE Operators via Spectral Filtering](https://mlsanity.com/paper/2601.17090/) (2026)
- [Two-Layer Linear Auto-Regressive Models Estimate Latent States](https://mlsanity.com/paper/2606.12691/) (2026)
- [Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models](https://mlsanity.com/paper/2608.05416/) (2026)

## Abstract (arXiv)

We study the fundamental problem of learning a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonlinear dynamical system that has finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This significantly generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.